Davis–Putnam algorithm


The Davis–Putnam algorithm was developed by Martin Davis and Hilary Putnam for checking the validity of a first-order logic formula using a resolution-based decision procedure for propositional logic. Since the set of valid first-order formulas is recursively enumerable but not recursive, there exists no general algorithm to solve this problem. Therefore, the Davis–Putnam algorithm only terminates on valid formulas. Today, the term "Davis–Putnam algorithm" is often used synonymously with the resolution-based propositional decision procedure that is actually only one of the steps of the original algorithm.

Overview

The procedure is based on Herbrand's theorem, which implies that an unsatisfiable formula has an unsatisfiable ground instance, and on the fact that a formula is valid if and only if its negation is unsatisfiable. Taken together, these facts imply that to prove the validity of φ it is enough to prove that a ground instance of ¬φ is unsatisfiable. If φ is not valid, then the search for an unsatisfiable ground instance will not terminate.
The procedure roughly consists of these three parts:
The last part is probably the most innovative one, and works as follows :
At each step, the intermediate formula generated is equisatisfiable, but possibly not equivalent, to the original formula. The resolution step leads to a worst-case exponential blow-up in the size of the formula.
The Davis–Putnam–Logemann–Loveland algorithm is a 1962 refinement of the propositional satisfiability step of the Davis–Putnam procedure which requires only a linear amount of memory in the worst case. It still forms the basis for today's most efficient complete SAT solvers.